Abstract

Pimsner introduced the C*-algebra OXgenerated by a Hilbert bimoduleXover a C*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, "(I)-freeness" and "(II)-freeness," stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251-268) and J. Cuntz (Invent. Math.63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of "Cuntz-Krieger bimodules." IfXsatisfies this condition the C*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz-Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.

title = "Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules",

abstract = "Pimsner introduced the C*-algebra OXgenerated by a Hilbert bimoduleXover a C*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, {"}(I)-freeness{"} and {"}(II)-freeness,{"} stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251-268) and J. Cuntz (Invent. Math.63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of {"}Cuntz-Krieger bimodules.{"} IfXsatisfies this condition the C*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz-Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.",

author = "Tsuyoshi Kajiwara and Claudia Pinzari and Yasuo Watatani",

year = "1998",

month = nov

day = "10",

doi = "10.1006/jfan.1998.3306",

language = "English",

volume = "159",

pages = "295--322",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules

AU - Kajiwara, Tsuyoshi

AU - Pinzari, Claudia

AU - Watatani, Yasuo

PY - 1998/11/10

Y1 - 1998/11/10

N2 - Pimsner introduced the C*-algebra OXgenerated by a Hilbert bimoduleXover a C*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, "(I)-freeness" and "(II)-freeness," stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251-268) and J. Cuntz (Invent. Math.63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of "Cuntz-Krieger bimodules." IfXsatisfies this condition the C*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz-Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.

AB - Pimsner introduced the C*-algebra OXgenerated by a Hilbert bimoduleXover a C*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, "(I)-freeness" and "(II)-freeness," stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251-268) and J. Cuntz (Invent. Math.63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of "Cuntz-Krieger bimodules." IfXsatisfies this condition the C*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz-Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.